Math 268     Introduction to Undergraduate Research     Spring 2011

Instructor: Sunil Chebolu


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Projects: I will outline some projects for you to work in small groups of size 4 or 5. These projects will be assigned to students on a first-come-first-serve basis. I will continuously update these projects and will also add links and references to useful resources. I find these projects incredibly beautiful, interesting, and fascinating. They are easily accessible (they take very little background) and will quickly take you on the road to research. The projects listed below need only basic techniques from elementary number theory, elementary Linear algebra, Euclidean geometry, and elementary probability theory. If you have not taken any of these courses, that is fine. I will teach you or provide you with the necessary background material as and when needed.


Research Teams and Abstracts

  1. The k-Numbers Game: Let k be a positive integer. We investigate the k-Numbers game, in which k numbers are arranged at the vertices of a regular polygon with k sides and the absolute value of the difference between adjacent numbers is taken until the numbers at all vertices become 0. We investigate symmetry in the game, upper bounds for the length of games, and the effect properties of numbers have on the length of games, and related questions.



  2. Pythagorean Triples and Integer Points on a Hyperboloid: We begin by studying the Pythagorean equation: x^2 + y^2 = z^2. We investigate what positive integer triples (x, y, z) will satisfy the Pythagorean equation and explore the patterns in the solutions that emerge. The natural extension of this work is to consider the equation of the Hyperboloid x^2 + y^2 = z^2 + k, where k is a fixed positive integer. We study the problem of finding integer points located on this Hyperboloid when k=2, i.e., solutions in positive integers. In particular, we investigate properties of the solutions and algorithms for generating all the integer points on the Hyperboloid.



  3. Pick's Theorem and its Applications Pick's theorem gives a nice formula to compute the area enclosed by a simple lattice polygon (a polygon whose vertices have integer coordinates). The area A of such a polygon P is given in terms of the number I of lattice points located in the interior of P and the number B of lattice points on the boundary of P as: A = I + B/2 - 1. We will investigate several consequences of this theorem.


Homework

  • Jan 25: Register for the undergraduate research syposium by Feb 7th.
  • Jan 20: Log on to blackboard to read the latex assignment. You have to upload your LaTeX file on Blackboard. Due: Feb 1st.
  • Jan 16: First LaTex exercise: Download Winedt to your laptops or desktops (use the Latex link above). Then download this SAMPLE FILE, and open it using Winedt.(Do not change the filename or its extension.) Finally, after you open this file in winedt, hit the Latex button on the top and you should be able to see the output in a .dvi file.
  • Jan 12: Be Wise, Generalise: I want you to generalise this to 3 dimensions: Call a cuboid (3-dimensional rectangle) cool if one if its 3 sides is integer length. Show that a cuboid that admits a tiling by cool subcuboids is itself cool.
  • Jan 11: In your research Journal, write a complete and precise proof of the following theorem which was discussed in class: A rectangle is called cool if either its length of width is of integer length. If R is a rectangle that admits a tiling by cool subrectangles, then R has to be cool.





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